BFGS optimization problem, setting up the equations

We are given the following unconstrained optimization problem:

$$\mathbf{x}^* = \arg \min_{\mathbf{x}} f(\mathbf{x})$$

where $f: \mathbb{R}^n \to \mathbb{R}$ is a differentiable objective function, and $\mathbf{x} \in \mathbb{R}^n$ is the vector of decision variables.

The first-order necessary conditions for optimality are given by:

$$\nabla f(\mathbf{x}^*) = 0$$

where $\nabla f(\mathbf{x})$ denotes the gradient of $f(\mathbf{x})$.

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